Tableau posets and the fake degrees of coinvariant algebra
Billey, Sara C. ; Konvalinka, Matjaž ; Swanson, Joshua P.
Vpeljemo dve delni urejenosti na standardnih Youngovih tabelah dane oblike, po analogiji s krepko in šibko Bruhatovo urejenostjo na permutacijah. Obe delno urejeni množici sta stopničasti glede na ...premaknjen majorski indeks. Obstoj take stopničaste strukture nam dovoljuje klasifikacijo vseh realizabilnih majorskih indeksov standardnih tabel poljubne ravne oblike in nekaterih poševnih oblik. Po izreku Lusztig-Stanleyja [R. P. Stanley, Bull. Am. Math. Soc., New Ser. 1, 475-511 (1979) Prop. 4.11] lahko to klasifikacijo interpretiramo kot ugotovitev, katere nerazcepne upodobitve simetrične grupe obstajajo v kateri homogeni komponenti pripadajoče koinvariantne algebre, s čimer izboljšamo nedavni rezultat tretjega avtorja za modularni majorski indeks. Naš pristop je, da identificiramo vzorce v standardnih tabelah, ki dovoljujejo mutacijo množice padcev na kontroliran način. Po rezultatih G. Lusztiga [Invent. Math. 43, 125-175 (1977] in J. R. Stembridgea [Pac. J. Math. 140, No. 2, 353-396 (1989)] lahko te rezultate razširimo na klasifikacijo vseh neničelnih lažnih stopenj koinvariantnih algber za končne kompleksne grupe zrcaljenj v neskončni družini Shephard-Toddovih grup.
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